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Properties

Label 164.5.d.f
Level $164$
Weight $5$
Character orbit 164.d
Analytic conductor $16.953$
Analytic rank $0$
Dimension $72$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [164,5,Mod(163,164)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(164, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("164.163");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 164 = 2^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 164.d (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(16.9526739458\)
Analytic rank: \(0\)
Dimension: \(72\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 6 q^{2} - 162 q^{4} - 8 q^{5} + 162 q^{8} + 1728 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q + 6 q^{2} - 162 q^{4} - 8 q^{5} + 162 q^{8} + 1728 q^{9} - 272 q^{10} - 2842 q^{16} - 298 q^{18} - 584 q^{20} + 280 q^{21} + 4264 q^{25} + 66 q^{32} + 3512 q^{33} - 11338 q^{36} + 5720 q^{37} - 4648 q^{40} - 7928 q^{41} - 10140 q^{42} - 9528 q^{45} - 5680 q^{46} + 6624 q^{49} + 554 q^{50} + 29864 q^{57} + 11712 q^{61} - 3936 q^{62} - 34554 q^{64} + 24852 q^{66} - 12166 q^{72} + 16632 q^{73} - 11120 q^{74} + 84456 q^{77} - 20496 q^{78} - 12104 q^{80} + 13408 q^{81} - 51114 q^{82} - 948 q^{84} - 20400 q^{86} + 64976 q^{90} - 2784 q^{92} + 2902 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
163.1 −3.71258 − 1.48887i −10.9113 11.5666 + 11.0551i 8.39889 40.5091 + 16.2454i −64.7429 −26.4823 − 58.2639i 38.0563 −31.1816 − 12.5048i
163.2 −3.71258 − 1.48887i 10.9113 11.5666 + 11.0551i 8.39889 −40.5091 − 16.2454i 64.7429 −26.4823 − 58.2639i 38.0563 −31.1816 − 12.5048i
163.3 −3.71258 + 1.48887i −10.9113 11.5666 − 11.0551i 8.39889 40.5091 − 16.2454i −64.7429 −26.4823 + 58.2639i 38.0563 −31.1816 + 12.5048i
163.4 −3.71258 + 1.48887i 10.9113 11.5666 − 11.0551i 8.39889 −40.5091 + 16.2454i 64.7429 −26.4823 + 58.2639i 38.0563 −31.1816 + 12.5048i
163.5 −3.64057 − 1.65719i −12.7376 10.5074 + 12.0662i −20.4399 46.3719 + 21.1085i 9.19140 −18.2571 − 61.3407i 81.2453 74.4128 + 33.8728i
163.6 −3.64057 − 1.65719i 12.7376 10.5074 + 12.0662i −20.4399 −46.3719 − 21.1085i −9.19140 −18.2571 − 61.3407i 81.2453 74.4128 + 33.8728i
163.7 −3.64057 + 1.65719i −12.7376 10.5074 − 12.0662i −20.4399 46.3719 − 21.1085i 9.19140 −18.2571 + 61.3407i 81.2453 74.4128 − 33.8728i
163.8 −3.64057 + 1.65719i 12.7376 10.5074 − 12.0662i −20.4399 −46.3719 + 21.1085i −9.19140 −18.2571 + 61.3407i 81.2453 74.4128 − 33.8728i
163.9 −3.33380 − 2.21038i −5.00494 6.22840 + 14.7379i 38.7793 16.6855 + 11.0629i 50.6913 11.8123 − 62.9005i −55.9505 −129.282 − 85.7171i
163.10 −3.33380 − 2.21038i 5.00494 6.22840 + 14.7379i 38.7793 −16.6855 − 11.0629i −50.6913 11.8123 − 62.9005i −55.9505 −129.282 − 85.7171i
163.11 −3.33380 + 2.21038i −5.00494 6.22840 − 14.7379i 38.7793 16.6855 − 11.0629i 50.6913 11.8123 + 62.9005i −55.9505 −129.282 + 85.7171i
163.12 −3.33380 + 2.21038i 5.00494 6.22840 − 14.7379i 38.7793 −16.6855 + 11.0629i −50.6913 11.8123 + 62.9005i −55.9505 −129.282 + 85.7171i
163.13 −2.77609 − 2.87981i −3.48340 −0.586657 + 15.9892i −32.0896 9.67024 + 10.0316i −52.0073 47.6747 − 42.6981i −68.8659 89.0837 + 92.4122i
163.14 −2.77609 − 2.87981i 3.48340 −0.586657 + 15.9892i −32.0896 −9.67024 − 10.0316i 52.0073 47.6747 − 42.6981i −68.8659 89.0837 + 92.4122i
163.15 −2.77609 + 2.87981i −3.48340 −0.586657 − 15.9892i −32.0896 9.67024 − 10.0316i −52.0073 47.6747 + 42.6981i −68.8659 89.0837 − 92.4122i
163.16 −2.77609 + 2.87981i 3.48340 −0.586657 − 15.9892i −32.0896 −9.67024 + 10.0316i 52.0073 47.6747 + 42.6981i −68.8659 89.0837 − 92.4122i
163.17 −2.41605 − 3.18790i −7.04202 −4.32538 + 15.4043i 7.90186 17.0139 + 22.4492i 68.9363 59.5575 − 23.4286i −31.4099 −19.0913 − 25.1903i
163.18 −2.41605 − 3.18790i 7.04202 −4.32538 + 15.4043i 7.90186 −17.0139 − 22.4492i −68.9363 59.5575 − 23.4286i −31.4099 −19.0913 − 25.1903i
163.19 −2.41605 + 3.18790i −7.04202 −4.32538 − 15.4043i 7.90186 17.0139 − 22.4492i 68.9363 59.5575 + 23.4286i −31.4099 −19.0913 + 25.1903i
163.20 −2.41605 + 3.18790i 7.04202 −4.32538 − 15.4043i 7.90186 −17.0139 + 22.4492i −68.9363 59.5575 + 23.4286i −31.4099 −19.0913 + 25.1903i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 163.72
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
41.b even 2 1 inner
164.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 164.5.d.f ✓ 72
4.b odd 2 1 inner 164.5.d.f ✓ 72
41.b even 2 1 inner 164.5.d.f ✓ 72
164.d odd 2 1 inner 164.5.d.f ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
164.5.d.f ✓ 72 1.a even 1 1 trivial
164.5.d.f ✓ 72 4.b odd 2 1 inner
164.5.d.f ✓ 72 41.b even 2 1 inner
164.5.d.f ✓ 72 164.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{36} - 1890 T_{3}^{34} + 1613788 T_{3}^{32} - 824581952 T_{3}^{30} + 281530979028 T_{3}^{28} + \cdots + 62\!\cdots\!08 \) acting on \(S_{5}^{\mathrm{new}}(164, [\chi])\). Copy content Toggle raw display